On the distribution of roots of polynomials in sectors. II
Copyright policy )On the distribution of roots of polynomials in sectors. II
Let \(P(X)=a_nX^n+\cdots+a_0\) (\(a_0,a_n\neq 0\)) be a complex polynomial. For \(0\leq\psi\leq\varphi\leq 2\pi\) denote by \(N_P(\psi,\varphi)\) the number of roots \(z\) of \(P\) satisfying \(\psi\leq\arg z\leq\varphi\), and put \[ E_P=E_P(\psi,\varphi)=\biggl| N_P(\psi,\varphi)-{(\psi-\varphi)n\over 2\pi} \biggr|. \] It was shown by \textit{P. Erdős} and \textit{P. Turán} [Ann. Math. (2) 51, 105-119 (1950; Zbl 0036.01501)] that if \(L(P)\) denotes the sum of absolute values of coefficients of \(P\), then \[ E_P<16\left(n\log\left({L(P)\over a_na_0}\right)\right)^{1/2}. \] \textit{T. Ganelius} [Ark. Mat. 3, 1-50 (1954; Zbl 0055.06905)] replaced the coefficient \(16\) by \(2.61\dots\). Later an upper bound for \(E_P\) involving Mahler's measure of \(P\) was proved by \textit{M. Mignotte} [Acta Arith. 54, 81-86 (1989; Zbl 0641.12003)]. The author considers the case \(n\geq 10\) and obtains (Theorem 1) a bound of the form \(E_P
Mahler's measure, discriminant, roots of polynomials in sectors, roots of polynomials, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), Polynomials in real and complex fields: location of zeros (algebraic theorems), complex polynomial
Mahler's measure, discriminant, roots of polynomials in sectors, roots of polynomials, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), Polynomials in real and complex fields: location of zeros (algebraic theorems), complex polynomial
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